Compute Differential Privacy level for any randomised algorithm

I have recently started learning differential privacy for my BTech project. I understand that it adds noise to the input stream based on a privacy level (say $\epsilon$) and a query function (say $f$), to provide privacy to the input dataset. The distribution parameters for the noisy signals are computed based on these things only.

Now suppose, we have a randomized algorithm that adds noise to the input dataset based on its corresponding privacy level (say $\pi$) and does not consider any query function during its computation. The algorithm just provides noise values based on its privacy formulation.

Now, if I choose a query function (say the same function $f$) and try to model the noisy signals of the randomized algorithm to find the differential privacy level ($\epsilon$) of the randomized algorithm, is it possible to establish a relationship between $\epsilon$ and $\pi$?